TensorSum

TensorSum
Computes a vector or operator from its tensor decomposition
Other toolboxes required	none
Related functions	FilterNormalForm OperatorSchmidtDecomposition SchmidtDecomposition
Function category	Basic operation

TensorSum is a function that computes a vector or operator from its tensor decomposition. It acts as an inverse of the OperatorSchmidtDecomposition and SchmidtDecomposition functions, but also works for multipartite decompositions.

Syntax

• TS = TensorSum(A1,A2,...)
• TS = TensorSum(S,A1,A2,...)

Argument descriptions

• A1,A2,...: Either matrices or cells containing matrices (and they should all be the same: either all matrices or all cells). If they are matrices, then the k-th column of each Ai will be tensored together for all k, and then the sum over k will be taken at the end. If they are cells, then the k-th element of each Ai will be tensored together for all k, and then the sum over k will be taken at the end.
• S (optional, default [1,1,...,1]): A vector of weights (such as Schmidt coefficients) that will be applied when summing the terms at the end of the computation.

Examples

Inverse of SchmidtDecomposition

This function acts as an inverse of the SchmidtDecomposition function:

>> vec = [1,2,3,4];
>> [s,u,v] = SchmidtDecomposition(vec);
>> TensorSum(s,u,v)
 
ans =
 
    1.0000
    2.0000
    3.0000
    4.0000

Inverse of OperatorSchmidtDecomposition

This function also acts as an inverse of the OperatorSchmidtDecomposition function:

>> X = reshape(1:16,4,4)'
 
X =
 
     1     2     3     4
     5     6     7     8
     9    10    11    12
    13    14    15    16
 
>> [s,U,V] = OperatorSchmidtDecomposition(X);
>> TensorSum(s,U,V)
 
ans =
 
    1.0000    2.0000    3.0000    4.0000
    5.0000    6.0000    7.0000    8.0000
    9.0000   10.0000   11.0000   12.0000
   13.0000   14.0000   15.0000   16.0000

Source code

Click on "expand" to the right to view the MATLAB source code for this function.

1. %%  TENSORSUM  Computes a vector or operator from its tensor decomposition

2. %   This function can be called in two ways:

3. %   TS = TensorSum(A1,A2,...)

4. %   TS = TensorSum(S,A1,A2,...)

5. %

6. %   Each term Ai is either a matrix or a cell containing matrices (and they

7. %   should all be the same: either all matrices or all cells). If they are

8. %   matrices, then the k-th column of each Ai will be tensored together for

9. %   all k, and then the sum over k will be taken at the end. If they are

10. %   cells, then the k-th element of each Ai will be tensored together for

11. %   all k, and then the sum over k will be taken at the end.

12. %

13. %   If S is provided, it is a vector of weights (such as Schmidt

14. %   coefficients) that will be applied when summing the terms at the end of

15. %   the computation.

16. %

17. %   This function acts as an inverse of the SchmidtDecomposition and

18. %   OperatorSchmidtDecomposition functions. For example, if [S,U,V] =

19. %   SchmidtDecomposition(VEC) then VEC = TensorSum(S,U,V).

20. %

21. %   URL: http://www.qetlab.com/TensorSum

23. %   requires: nothing

24. %   author: Nathaniel Johnston (nathaniel@njohnston.ca)

25. %   package: QETLAB

26. %   last updated: December 19, 2014

28. function ts = TensorSum(a1,a2,varargin)

30. % Determine the number of terms in the tensor decomposition.

31. if(iscell(a2))

32.     len = length(a2);

33. else

34.     len = size(a2,2);

35. end

37. % Determine whether or not a vector of coefficients was provided, then read

38. % in the input arguments.

39. if(iscell(a1) || min(size(a1)) > 1 || len == 1)

40.     s = ones(len,1);

41.     A = [{a1} {a2} varargin];

42. else

43.     s = a1;

44.     A = [{a2} varargin];

45. end

47. % Get some more preliminary values (dimensions, etc.) and then convert

48. % everything into cell matrices (instead of matrices of column vectors) if

49. % necessary.

50. num_parties = length(A);

51. dim = zeros(2,num_parties);

52. for j = 1:num_parties

53.     if(~iscell(A{j}))

54.         A{j} = mat2cell(A{j},size(A{j},1),ones(1,size(A{j},2)));

55.     end

56.     dim(:,j) = size(A{j}{1}).';

57. end

59. % Finally, actually compute the tensor sum.

60. ts = sparse(prod(dim(1,:)),prod(dim(2,:)));

61. for j = 1:len

62.     ts_tmp = A{1}{j};

63.     for k = 2:num_parties

64.         ts_tmp = kron(ts_tmp,A{k}{j});

65.     end

66.     ts = ts + s(j)*ts_tmp;

67. end